Existence and convergence theorems concerning common fixed points of nonlinear semigroups of weak contractions
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Journal of Fixed Point Theory and Applications
Existence and convergence theorems concerning common fixed points of nonlinear semigroups of weak contractions Thitima Kesahorm and Wutiphol Sintunavarat Abstract. Nowadays, the usage of the existence and convergence results of a common fixed point for nonlinear semigroups of various kinds of selfmappings is applied to several problems in mathematics and another branches. Surprisingly, nobody considered the study on the nonlinear semigroups of weak contraction self-mappings in the sense of Berinde. The purpose of this paper is to attempt to complete this direction and so we investigate the existence of the common fixed point for nonlinear semigroups of weak contraction self-mappings in the sense of Berinde on a bounded closed convex subset of a real Banach space with uniformly normal structure. More precisely, we prove the convergence theorem for the common fixed point of a weak contraction semigroups. An illustrative example is presented. Mathematics Subject Classification. 47H06, 47J20, 47J25. Keywords. Common fixed point, weak contraction mapping, nonlinear semigroup, convergence theorem.
1. Introduction Fixed point theory is one of the most important tools in several areas of mathematics and other sciences. In the study of fixed point theory, there are two elementary questions as follows: Q1 : What additional conditions must be added to assure that a mapping has at least one fixed point ? Q2 : If a mapping has a fixed point, how one can approximate such a fixed point ? Many famous fixed point theorems have provided an answer to the above questions and have been investigated by several authors such as Banach [2], Caristi [7], Krasnoselskii [18], and so on (see [16,19,23], and the references therein). As is well known, fixed point iterative methods are important methods for constructing fixed points of various contractive type mappings (see, 0123456789().: V,-vol
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for example, [14,25,29]). Moreover, fixed point iterative methods can be applied successfully to a wide range of contexts in various topics, for example, differential equations, integral equations, matrix equations, convex minimization, split feasibility, image recovery, and signal processing problems (see, for example, [5,30,31]). In 2004, Berinde [3] introduced the concept of a new contractive condition as follows: Definition 1.1. Let (X, d) be a metric space. A mapping T : X → X is called a weak contraction mapping in the sense of Berinde if there are constants δ ∈ [0, 1) and L ≥ 0 such that d(T x, T y) ≤ δd(x, y) + Ld(y, T x)
(1.1)
for all x, y ∈ X. Also, Berinde [3] proved the following two fixed point theorems: Theorem 1.2. [3] Let (X, d) be a complete metric space and T : X → X be a weak contraction mapping, i.e., a mapping satisfying the condition (1.1) with δ ∈ [0, 1) and some L ≥ 0. Then, T has a fixed point. Moreover, for each x0 ∈ X, the Picard iteration {xn }, which is defined by xn+1 = T xn for all n ∈ N, converges to a fixed point of T . Theorem 1.3. [3] Le
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